Inverse Trig Functions 6.1 JMerrill, 2007 Revised 2009.
-
Upload
ada-fleming -
Category
Documents
-
view
220 -
download
3
Transcript of Inverse Trig Functions 6.1 JMerrill, 2007 Revised 2009.
![Page 1: Inverse Trig Functions 6.1 JMerrill, 2007 Revised 2009.](https://reader036.fdocuments.us/reader036/viewer/2022062722/56649f285503460f94c40f23/html5/thumbnails/1.jpg)
Inverse Trig FunctionsInverse Trig Functions6.16.1
Inverse Trig FunctionsInverse Trig Functions6.16.1
JMerrill, 2007JMerrill, 2007
Revised 2009Revised 2009
![Page 2: Inverse Trig Functions 6.1 JMerrill, 2007 Revised 2009.](https://reader036.fdocuments.us/reader036/viewer/2022062722/56649f285503460f94c40f23/html5/thumbnails/2.jpg)
Recall• From College Algebra, we know
that for a function to have an inverse that is a function, it must be one-to-one—it must pass the Horizontal Line Test.
![Page 3: Inverse Trig Functions 6.1 JMerrill, 2007 Revised 2009.](https://reader036.fdocuments.us/reader036/viewer/2022062722/56649f285503460f94c40f23/html5/thumbnails/3.jpg)
Sine Wave• From looking at a sine wave, it is
obvious that it does not pass the Horizontal Line Test.
![Page 4: Inverse Trig Functions 6.1 JMerrill, 2007 Revised 2009.](https://reader036.fdocuments.us/reader036/viewer/2022062722/56649f285503460f94c40f23/html5/thumbnails/4.jpg)
Sine Wave• In order to pass the Horizontal
Line Test (so that sin x has an inverse that is a function), we must restrict the domain.
• We restrict it to ,
2 2
![Page 5: Inverse Trig Functions 6.1 JMerrill, 2007 Revised 2009.](https://reader036.fdocuments.us/reader036/viewer/2022062722/56649f285503460f94c40f23/html5/thumbnails/5.jpg)
Sine Wave• Quadrant IV is • Quadrant I is • Answers must be in one of those
two quadrants or the answer doesn’t exist.
,02
0,2
![Page 6: Inverse Trig Functions 6.1 JMerrill, 2007 Revised 2009.](https://reader036.fdocuments.us/reader036/viewer/2022062722/56649f285503460f94c40f23/html5/thumbnails/6.jpg)
Sine Wave• How do we draw inverse
functions?• Switch the x’s and y’s!Switching the x’s and y’s also
means switching the axis!
![Page 7: Inverse Trig Functions 6.1 JMerrill, 2007 Revised 2009.](https://reader036.fdocuments.us/reader036/viewer/2022062722/56649f285503460f94c40f23/html5/thumbnails/7.jpg)
Sine Wave• Domain/range of restricted wave?• Domain/range of inverse?
: ,2 2
: 1,1
D
R
: 1,1
: ,2 2
D
R
![Page 8: Inverse Trig Functions 6.1 JMerrill, 2007 Revised 2009.](https://reader036.fdocuments.us/reader036/viewer/2022062722/56649f285503460f94c40f23/html5/thumbnails/8.jpg)
Inverse Notation• y = arcsin x or y = sin-1 x
• Both mean the same thing. They mean that you’re looking for the angle (y) where sin y = x.
![Page 9: Inverse Trig Functions 6.1 JMerrill, 2007 Revised 2009.](https://reader036.fdocuments.us/reader036/viewer/2022062722/56649f285503460f94c40f23/html5/thumbnails/9.jpg)
Evaluating Inverse Functions
• Find the exact value of:• Arcsin ½
– This means at what angle is the sin = ½ ?
– π/6– 5π/6 has the same answer, but falls
in QIII, so it is not correct.
![Page 10: Inverse Trig Functions 6.1 JMerrill, 2007 Revised 2009.](https://reader036.fdocuments.us/reader036/viewer/2022062722/56649f285503460f94c40f23/html5/thumbnails/10.jpg)
Calculator• When looking for an inverse answer
on the calculator, use the 2nd key first, then hit sin, cos, or tan.
• When looking for an angle always hit the 2nd key first.
• Last example: Degree mode, 2nd, sin, .5 = 30.
![Page 11: Inverse Trig Functions 6.1 JMerrill, 2007 Revised 2009.](https://reader036.fdocuments.us/reader036/viewer/2022062722/56649f285503460f94c40f23/html5/thumbnails/11.jpg)
Evaluating Inverse Functions
• Find the value of:• sin-1 2
– This means at what angle is the sin = 2 ?
– What does your calculator read? Why?
– 2 falls outside the range of a sine wave and outside the domain of the inverse sine wave
![Page 12: Inverse Trig Functions 6.1 JMerrill, 2007 Revised 2009.](https://reader036.fdocuments.us/reader036/viewer/2022062722/56649f285503460f94c40f23/html5/thumbnails/12.jpg)
Cosine Wave
![Page 13: Inverse Trig Functions 6.1 JMerrill, 2007 Revised 2009.](https://reader036.fdocuments.us/reader036/viewer/2022062722/56649f285503460f94c40f23/html5/thumbnails/13.jpg)
Cosine Wave• We must restrict the domain• Now the inverse
: 0,
: 1,1
D
R
: 1,1
: 0,
D
R
![Page 14: Inverse Trig Functions 6.1 JMerrill, 2007 Revised 2009.](https://reader036.fdocuments.us/reader036/viewer/2022062722/56649f285503460f94c40f23/html5/thumbnails/14.jpg)
Cosine Wave• Quadrant I is • Quadrant II is • Answers must be in one of those
two quadrants or the answer doesn’t exist.
0,2
,2
![Page 15: Inverse Trig Functions 6.1 JMerrill, 2007 Revised 2009.](https://reader036.fdocuments.us/reader036/viewer/2022062722/56649f285503460f94c40f23/html5/thumbnails/15.jpg)
Tangent Wave
![Page 16: Inverse Trig Functions 6.1 JMerrill, 2007 Revised 2009.](https://reader036.fdocuments.us/reader036/viewer/2022062722/56649f285503460f94c40f23/html5/thumbnails/16.jpg)
Tangent Wave• We must restrict the domain• Now the inverse
![Page 17: Inverse Trig Functions 6.1 JMerrill, 2007 Revised 2009.](https://reader036.fdocuments.us/reader036/viewer/2022062722/56649f285503460f94c40f23/html5/thumbnails/17.jpg)
Graphing Utility: Graph the following inverse functions.
a. y = arcsin x
b. y = arccos x
c. y = arctan x
–1.5 1.5
–
–1.5 1.5
2
–
–3 3
–
Set calculator to radian mode.
![Page 18: Inverse Trig Functions 6.1 JMerrill, 2007 Revised 2009.](https://reader036.fdocuments.us/reader036/viewer/2022062722/56649f285503460f94c40f23/html5/thumbnails/18.jpg)
Graphing Utility: Approximate the value of each expression.
a. cos–1 0.75 b. arcsin 0.19
c. arctan 1.32 d. arcsin 2.5
Set calculator to radian mode.
![Page 19: Inverse Trig Functions 6.1 JMerrill, 2007 Revised 2009.](https://reader036.fdocuments.us/reader036/viewer/2022062722/56649f285503460f94c40f23/html5/thumbnails/19.jpg)
Composition of Functions
• Find the exact value of•
• Where is the sine =• Replace the parenthesis in the
original problem with that answer• Now solve
1 2sin sin
2
22 4
sin4 2
2
![Page 20: Inverse Trig Functions 6.1 JMerrill, 2007 Revised 2009.](https://reader036.fdocuments.us/reader036/viewer/2022062722/56649f285503460f94c40f23/html5/thumbnails/20.jpg)
Example• Find the exact value of
• The sine angles must be in QI or QIV, so we must use the reference angle
•
1 3sin sin
4
2sin
4 2
4
1 13sin sin sin sin
4 4
1 2sin
2
4
![Page 21: Inverse Trig Functions 6.1 JMerrill, 2007 Revised 2009.](https://reader036.fdocuments.us/reader036/viewer/2022062722/56649f285503460f94c40f23/html5/thumbnails/21.jpg)
Example• Find tan(arctan(-5))
-5• Find
• If the words are the same and the inverse function is inside the parenthesis, the answer is already given!
1 1cos cos
2
12
![Page 22: Inverse Trig Functions 6.1 JMerrill, 2007 Revised 2009.](https://reader036.fdocuments.us/reader036/viewer/2022062722/56649f285503460f94c40f23/html5/thumbnails/22.jpg)
Example• Find the exact value of• Steps:• Draw a triangle using only the
info inside the parentheses.• Now use your x, y, r’s
to answer the outside term
2tan arccos
3
2
3 5
yt n
52
ax
![Page 23: Inverse Trig Functions 6.1 JMerrill, 2007 Revised 2009.](https://reader036.fdocuments.us/reader036/viewer/2022062722/56649f285503460f94c40f23/html5/thumbnails/23.jpg)
Last Example• Find the exact value of• Cos is negative in QII and III, but
the inverse is restricted to QII.
1 7tan cos
12
-7
1295ytan
x957
![Page 24: Inverse Trig Functions 6.1 JMerrill, 2007 Revised 2009.](https://reader036.fdocuments.us/reader036/viewer/2022062722/56649f285503460f94c40f23/html5/thumbnails/24.jpg)
You Do• Find the exact value of
1 3tan sin
7
3 1020